How to attenuate an area light so that it falls off more rapidly than Quadratically
Here's a setup to get any falloff power - Constant, Linear, Quadratic, Cubic, etc. and everything inbetween :

Control the overall strength of the light by changing the Strength of the Light Falloff node. Changing the 'Power' changes the falloff with 0 being 'Constant' (ie, irrespective of distance), 1 being Linear falloff, 2 being Quadratic falloff, 3 being Cubic falloff, 4 to falloff to the power of 4, etc. You can adjust the power to any value including, say, 2.5 to be somewhere between Quadratic and Cubic.

EDIT : To get a bit more control over the effect close to the light source you add an extra Maths node to add an offset. This allows you to limit the lower end of the response by making so that it's as if the light is never closer than the specified amount. eg, setting to 0.5 the light will never be in the extreme range.

Another option would be to swap the 'Add' to, say, a Maximum. This way any distance below the set value will be treated as that value (and so not 'blow out') but it will still behave as the required falloff above that threshold.